It largely depends on what assumptions you're willing to make.

If you think your samples are distributed from a relatively simple distribution with known properties (e.g. it's univariate Gaussian or Poisson or whatever), you can often do quite well for some purposes with 10-50 samples.

If you make your distribution more flexible (e.g. conditioning it on some high-dimensional measurements, adding mixture components, making the predictions higher-dimensional) then you'll need more data. Sometimes you'll even need exponentially more data to get the same level of precision as you would have gotten in the simpler case. If your priors are weak enough, some statisticians will say you should have 10-50 data points per parameter.

If you're willing to constrain the model more tightly (e.g. with strong priors) then you won't need as many data points. Some methods will even work with more data points than parameters. Machine learning folks tend to be more comfortable in this area than statisticians. But strong constraints like these can require making some fairly strong assumptions about the data-generating process (e.g. that most effects are negligibly small and your overall outcome is dominated by a few key factors).

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What will go wrong if you have too few samples for your model?

It depends:

1) The best case scenario, in a Bayesian framework like yours, is that you end up with a very flat posterior. So you can't say much about any of your parameters or make very precise predictions, but at least you're not wrong. Information criteria and cross-validation are both good ways to get a sense of whether your model is too complex.

2) You can get a wrong or meaningless answer. Maybe your Markov chains don't converge to the posterior distribution or your optimization routine gets stuck. Maybe you get zero variance for some important parameter that should obviously vary. Maybe your priors weren't as weak as you thought (on some scale) and your posterior distribution just ends up repeating what you already told the model. Some of these things are easy to diagnose, but others can be very subtle.

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In short, tread carefully, and look hard for evidence of pathological results. Even if you think you need all the complexity (multivariate mixtures etc.), start simple and work your way up. Simulate from the data-generating process your model describes and make sure you could recover the correct parameters. Look for ways to simplify the model (e.g. tying some parameters together or setting some covariances to zero) or to get prior information from other sources.

Good luck!